| Summary: | <p><strong>Purpose: </strong>There is growing interest about biological effective dose (BED) and its application in treatment plan evaluation due to its stronger correlation with treatment outcome. An approximate biological effective dose (BED<sub>A</sub>) equation was introduced in order to simplify BED calculations by treatment planning systems in multi-phase treatments. The purpose of this work is to reveal its mathematical properties relative to the true, multi-phase BED (BED<sub>T</sub>) equation.</p><p><strong>Methods</strong>: The BED<sub>T</sub> equation was derived and used to reveal the mathematical properties of BED<sub>A</sub>. MATLAB (MathWorks, Natick, MA) was used to simulate and analyze common and extreme clinical multi-phase cases. In those cases, percent error and Bland-Altman analysis were used to study the significance of the inaccuracies of BED<sub>A</sub> for different combinations of total doses, numbers of fractions, doses per fractions and α/β values. All the calculations were performed on a voxel-basis in order to study how dose distributions would affect the accuracy of BED<sub>A</sub>.</p><p><strong>Results: </strong>When the voxel dose-per-fractions (DPF) delivered by both phases are equal, BED<sub>A</sub> and BED<sub>T</sub> are equal (0% error). In heterogeneous dose distributions, which significantly vary between the phases, there are fewer occurrences of equal DPFs and hence the imprecision of BED<sub>A</sub> is greater. It was shown that as the α/β ratio increased the accuracy of BED<sub>A</sub> would improve. Examining twenty-four cases, it was shown that the range of DPF ratios for 3% P<sub>error</sub> varied from 0.32 to 7.50Gy, whereas for P<sub>error</sub> of 1% the range varied from 0.50 to 2.96Gy.</p><p><strong>Conclusion</strong>: The DPF between the different phases should be equal in order to render BED<sub>A</sub> accurate. OARs typically receive heterogeneous dose distributions hence the probability of equal DPFs is low. Consequently, the BED<sub>A</sub> equation should only be used for targets or OARs that receive uniform or very similar dose distributions by the different treatment phases.</p><p><strong>---------------------------</strong></p><p><strong>Cite this article as</strong>: Kauweloa KI, Gutierrez AN, Bergamo A, Stathakis S, Papaniko-laou N, Mavroidis P. Mathematical analysis of approximate biological effective dose (BED) calculation for multi-phase radiotherapy treatment plans. Int J Cancer Ther Oncol 2014; 2(2):020226. <strong>DOI: 10.14319/ijcto.0202.26</strong></p>
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